Lecture 8

Distributions

Normal distributions and the sampling distribution



Dr Lincoln Colling

14 Nov 2022


Psychology as a Science

Plan for today

Today we’ll learn about the sampling distribution

But before we can do that we need to know what distributions are, where they come from, and how to describe them

  • The binomial distribution

  • The normal distribution

    • Processes that produce normal distributions

    • Process that don’t produce normal distributions

    • Describing normal distributions

    • Describing departures from the normal distributions

  • Distributions and samples

    • The Central Limit Theorem

  • The Standard Error of the Mean

The Binomial Distributions

  • The binomial distribution is one of the simplest distribution you’ll come across

  • To see where it comes from, we’ll just build one!

  • We can build one by flipping a coin (multiple times) and counting up the number of heads that we get

Figure 1: Possible sequences after coin flips

Figure 2: Distribution of number of heads after coin flips

  • In Figure 1 we can see the possible sequences of events that can happen if we flip a coin (⚈ = heads and ⚆ = tails) Figure 2 look very interesting at the moment.

  • In Figure 2 we just count up the number of sequences that lead to 0 heads, 1 head, 2 heads, etc

  • As we flip more coins the distribution of number of heads takes on a characteristic shape

  • This is the binomial distribution

The binomial distribution

  • The binomial distribution is just an idealised representation of the process that generates sequences of heads and tails when we flip a coin

    • Or any other process that gives rise to binary data

  • It’s an idealisation but natural processes do give rise to binomial distribution

  • In the bean machine (Figure 3) balls fall from the top and bounce off pegs as they fall

    • Balls can bounce one of two directions (left or right; binary outcome)

  • Most of the balls collect near the middle, and fewer balls are found at the edges

Figure 3: Example of the bean machine

The normal distribution

Flipping coins might seem a long way off anything you might want to study in psychology, but the shape of the binomial distribution might be familiar to you

  • The binomial distribution has a shape that is similar to the normal distribution

But there are a few key differences:

  1. The binomial distribution is bounded at 0 and n (number of coins)

    • The normal distribution can range from \(+\infty\) to \(-\infty\)

  2. The binomial distribution is discrete (0, 1, 2, 3 etc, but no 2.5)

    • The normal distribution is continuous

The normal distribution is a mathematical abstraction, but we can use it as model of real-life populations that are produced by certain kinds of natural processes

Processes that produce normal distributions

To see how a natural process can give rise to a normal distribution, let’s play a board game!

There’s only 1 rule: You roll the dice n times (number of rounds), add up all the values, and move than many spaces. That is your score

  • We can play any number of rounds

  • And we’ll play with friends, because you can’t get a distribution of scores if you play by yourself!

If we have enough players who play enough rounds then the distribution of scores across all the players will take on a characteristic shape

Figure 4: Distribution of players’ position from the starting point

Processes that produce normal distributions

  • A players score on the dice game is determined by adding up the values of each roll

  • So after each roll their score can increase by some amount

The dice game might look artificial, but it maybe isn’t that different to some natural processes

For example, developmental processes might look pretty similar to the dice game

Think about height:

  • At each point in time some value can be added (growth) or a person’s current height

  • So if we looked at the distribution of heights in the population then we might find something that looks similar to a normal distribution

A key factor that results in the normal distribution shape is this adding up of values

Processes that don’t produce normal distributions

Let’s change the rules of the game

  • Instead of adding up the value of each roll, we’ll multiply them ( e.g., roll a 1, 2, and 4 and your score is 8)

  • The distribution is skewed with most player having low scores and a few players have very high scores

  • Can you think of a process that operates like this in the real world?

    • How about interest or returns on investments?

    • Maybe this explains the shape of real world wealth distributions…